The characteristic-rank conjecture for oriented Grassmannians

Let $\widetilde{\operatorname{Gr}}_k(n)$ be the oriented Grassmannian and let $S\to\widetilde{\operatorname{Gr}}_k(n)$ be its tautological bundle. Write $\mathrm{crk}(S)$ for the characteristic rank of $S$. For integers satisfying

$$5\leq k\leq 2^{t-1}<n\leq 2^t,\qquad t\geq 5,$$

Characteristic-rank conjecture. The characteristic rank is

$$\mathrm{crk}(S)=\min\bigl(2^t-2,\,k(n-2^{t-1})+2^{t-1}-2\bigr).$$

The theorem immediately preceding the conjecture establishes the upper bound $\mathrm{crk}(S)\leq 2^t-2$ when $n=2^t$ and $2^t-5\geq k\geq5$, producing a nonzero anomalous class. The conjecture predicts the exact characteristic rank throughout the displayed range, while the general equality remains open.